Average Error: 28.6 → 0.2
Time: 3.7s
Precision: 64
\[\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2}\]
↓
\[0.5 \cdot \left(\left(y + \frac{x}{\frac{y}{x}}\right) - z \cdot \frac{z}{y}\right)\]
\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2}
↓
0.5 \cdot \left(\left(y + \frac{x}{\frac{y}{x}}\right) - z \cdot \frac{z}{y}\right)
double f(double x, double y, double z) {
        double r656608 = x;
        double r656609 = r656608 * r656608;
        double r656610 = y;
        double r656611 = r656610 * r656610;
        double r656612 = r656609 + r656611;
        double r656613 = z;
        double r656614 = r656613 * r656613;
        double r656615 = r656612 - r656614;
        double r656616 = 2.0;
        double r656617 = r656610 * r656616;
        double r656618 = r656615 / r656617;
        return r656618;
}
↓
double f(double x, double y, double z) {
        double r656619 = 0.5;
        double r656620 = y;
        double r656621 = x;
        double r656622 = r656620 / r656621;
        double r656623 = r656621 / r656622;
        double r656624 = r656620 + r656623;
        double r656625 = z;
        double r656626 = r656625 / r656620;
        double r656627 = r656625 * r656626;
        double r656628 = r656624 - r656627;
        double r656629 = r656619 * r656628;
        return r656629;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

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Results

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Target

Original28.6
Target0.2
Herbie0.2
\[y \cdot 0.5 - \left(\frac{0.5}{y} \cdot \left(z + x\right)\right) \cdot \left(z - x\right)\]

Derivation

  1. Initial program 28.6

    \[\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2}\]

  2. Taylor expanded around 0 12.5

    \[\leadsto \color{blue}{\left(0.5 \cdot y + 0.5 \cdot \frac{{x}^{2}}{y}\right) - 0.5 \cdot \frac{{z}^{2}}{y}}\]

  3. Simplified12.5

    \[\leadsto \color{blue}{0.5 \cdot \left(\left(y + \frac{{x}^{2}}{y}\right) - \frac{{z}^{2}}{y}\right)}\]
  4. Using strategy rm

  5. Applied *-un-lft-identity12.5

    \[\leadsto 0.5 \cdot \left(\left(y + \frac{{x}^{2}}{y}\right) - \frac{{z}^{2}}{\color{blue}{1 \cdot y}}\right)\]

  6. Applied add-sqr-sqrt38.7

    \[\leadsto 0.5 \cdot \left(\left(y + \frac{{x}^{2}}{y}\right) - \frac{{\color{blue}{\left(\sqrt{z} \cdot \sqrt{z}\right)}}^{2}}{1 \cdot y}\right)\]

  7. Applied unpow-prod-down38.7

    \[\leadsto 0.5 \cdot \left(\left(y + \frac{{x}^{2}}{y}\right) - \frac{\color{blue}{{\left(\sqrt{z}\right)}^{2} \cdot {\left(\sqrt{z}\right)}^{2}}}{1 \cdot y}\right)\]

  8. Applied times-frac36.0

    \[\leadsto 0.5 \cdot \left(\left(y + \frac{{x}^{2}}{y}\right) - \color{blue}{\frac{{\left(\sqrt{z}\right)}^{2}}{1} \cdot \frac{{\left(\sqrt{z}\right)}^{2}}{y}}\right)\]

  9. Simplified36.0

    \[\leadsto 0.5 \cdot \left(\left(y + \frac{{x}^{2}}{y}\right) - \color{blue}{z} \cdot \frac{{\left(\sqrt{z}\right)}^{2}}{y}\right)\]

  10. Simplified6.7

    \[\leadsto 0.5 \cdot \left(\left(y + \frac{{x}^{2}}{y}\right) - z \cdot \color{blue}{\frac{z}{y}}\right)\]
  11. Using strategy rm

  12. Applied unpow26.7

    \[\leadsto 0.5 \cdot \left(\left(y + \frac{\color{blue}{x \cdot x}}{y}\right) - z \cdot \frac{z}{y}\right)\]

  13. Applied associate-/l*0.2

    \[\leadsto 0.5 \cdot \left(\left(y + \color{blue}{\frac{x}{\frac{y}{x}}}\right) - z \cdot \frac{z}{y}\right)\]

  14. Final simplification0.2

    \[\leadsto 0.5 \cdot \left(\left(y + \frac{x}{\frac{y}{x}}\right) - z \cdot \frac{z}{y}\right)\]

Reproduce

herbie shell --seed 2020001 +o rules:numerics
(FPCore (x y z)
  :name "Diagrams.TwoD.Apollonian:initialConfig from diagrams-contrib-1.3.0.5, A"
  :precision binary64

  :herbie-target
  (- (* y 0.5) (* (* (/ 0.5 y) (+ z x)) (- z x)))

  (/ (- (+ (* x x) (* y y)) (* z z)) (* y 2)))